is a model one for the study of the \(L^p\) -maximal regularity of (REF ).
We refer the reader to [1]} or [2]} for the background on the \(L^p\) -maximal regularity the abstract Cauchy problem (REF ).
In this context, we point out that the kernels of Fourier multipliers involved in the study of maximal regularity have supports in \(\mathbb {R}_+\) , which allows considering a larger class of Banach function spaces \(\Phi \)  over \((\mathbb {R}_+, \mathrm {d}t)\) , and consequently proving the existence of strong solutions of (REF ) for a larger class of functions \(f\) ; see [3]} and the references therein.
